Prove that the sequence
lim n-->infinity 1/n(1 + 1/2 + ...+ 1/n) = 0
is
(i) monotone
(ii) bounded
(iii) find its limits
How can i proceed to prove that it is monotone and bounded???
Help me please!
Thank You!
If 1 + 1/2 + ... + 1/n is in the denominator, then the denominator itself is increasing (at each step the sum gets bigger, and it gets multiplier by a greater number). This implies that the sequence is decreasing. It is bounded below since all its terms are positive. And it will converge to 0, since it is smaller than 1/n, which converges to 0.